Chapter 5

Chapter 5

Knowledge Representation & Reasoning (Part 1)

Propositional Logic

King Saud UniversityCollege of Computer and Information Sciences Information Technology DepartmentIT422 - Intelligent systems

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Knowledge Representation & Reasoning

Introduction

How can we formalize our knowledge about the world so that:

• We can reason about it?

• We can do sound inference?

• We can prove things?

• We can plan actions?

• We can understand and explain things?

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Introduction

Objectives of knowledge representation and reasoning are:

• form representations of the world.

• use a process of inference to derive new representations about the world.

• use these new representations to deduce what to do.

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Introduction

Some definitions:• Knowledge base: set of sentences. Each sentence is expressed

in a language called a knowledge representation language.

• Sentence: a sentence represents some assertion about the world.

• Inference: Process of deriving new sentences from old ones.

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Introduction

Declarative vs. procedural approach:

• Declarative approach is an approach to system building that consists in expressing the knowledge of the environment in the form of sentences using a representation language.

• Procedural approach encodes desired behaviors directly as a program code.

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• Example: The Wumpus world

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About The Wumpus• The Wumpus world was first written as a computer game in

the 70ies;• the wumpus world is a cave consisting of rooms connected by

passageways;• the terrible wumpus is a beast that eats anyone who enters its

room;• the wumpus can be shot by an agent, but the agent has only

one arrow;• some rooms contain bottomless pits that will trap anyone

who enters it;• the agent is in this cave to look for gold.

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Environment• Squares adjacent to wumpus are

smelly.• Squares adjacent to pit are

breezy.• Glitter if and only if gold is in the

same square.• Shooting kills the wumpus if you

are facing it.• Shooting uses up the only arrow.• Grabbing picks up the gold if in

the same square.• Releasing drops the gold in the

same square.

Goals: Get gold back to the start without entering wumpus square.

Percepts: Breeze, Glitter, Smell.

Actions: Left turn, Right turn, Forward, Grab, Release, Shoot.

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The Wumpus world

• Is the world deterministic?Yes: outcomes are exactly specified.

• Is the world fully accessible?No: only local perception of square you are in.

• Is the world static?Yes: Wumpus and Pits do not move.

• Is the world discrete?Yes.

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A

Exploring Wumpus World

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okA

Ok because:

Haven’t fallen into a pit.

Haven’t been eaten by a Wumpus.


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OK

OK OK

OK since

no Stench, no Breeze,

neighbors are safe (OK).

A


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We move and smell a stench.

A


OKstench

OK OK

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W?

OKstench

W?

OK OK

We can infer the following.

Note: square (1,1) remains OK. A


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W?

OKstench

W?

OKOKbreeze

A

Move and feel a breeze

What can we conclude?


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W?

OKstench

P?W?

OKOKbreeze

P?

And what about the other P? and W? squares

But, can the 2,2 square really have either a Wumpus or a pit? ANO!


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W

OKstench

P?W?

OKOKbreeze

PA


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W OK

OKstench

OK OK

OKOKbreeze

P

A


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W

OKStench

OK

OKOKBreeze

P

A

A…And the exploration continues onward until the gold is found. …


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Breeze in (1,2) and (2,1)

no safe actions.

Assuming pits uniformly distributed, (2,2) is most likely to have a pit.

A tight spot

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W?

W?

Smell in (1,1) cannot move.

Can use a strategy of coercion:– shoot straight ahead;– wumpus was there

dead safe.– wumpus wasn't there

safe.

Another tight spot

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Fundamental property of logical reasoning:

• In each case where the agent draws a conclusion from the available information, that conclusion is guaranteed to be correct if the available information is correct.

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Fundamental concepts of logical representation:• Logics are formal languages for representing information such that

conclusions can be drawn.• Each sentence is defined by a syntax and a semantic.• Syntax defines the sentences in the language. It specifies well formed

sentences.• Semantics define the ``meaning'' of sentences;• i.e., in logic it defines the truth of a sentence in a possible world.

For example, in the language of arithmetic:x + 2 y is a sentence.x + y > is not a sentence.x + 2 y is true if the number x+2 is no less than the number y.x + 2 y is true in a world where x = 7, y =1.x + 2 y is false in a world where x = 0, y= 6.

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Fundamental concepts of logical representation:• Model: This word is used instead of “possible world” for the sake of precision.

m is a model of a sentence α means α is true in model mM(α) means the set of all models of α

• Definition: A model is a mathematical abstraction that simply fixes the truth or falsehood of every relevant sentence.

• Example: x is the number of men and y is the number of women sitting at a table playing bridge.

x+ y = 4 is a sentence which is true when the total number is four.

In this case the Model is a possible assignment of numbers to the variables x and y. Each assignment fixes the truth of any sentence whose variables are x and y.

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Potential models of the Wumpus world

Following the wumpus-world rules, in which models the KB corresponding to the observations of nothing in [1,1] and breeze in [2,1] is true?

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Potential models of the Wumpus world

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Fundamental concepts of logical representation:

• Entailment: Logical reasoning requires the relation of logical entailment between sentences. « a sentence follows ⇒logically from another sentence ».

Mathematical notation: α β (α entails the sentence β)╞

• Formal definition: α β if and only if ╞ in every model in which α is true, β is also true. (The truth of α is contained in the truth of β).

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Entailment• α β if and only if ╞ in every model in which α is true, β is also true. (The

truth of α is contained in the truth of β).

Models

β = T

α = T

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Entailment

Logical Representation

World

SentencesKB

FactsSem

antics

Sentences

Semantics

FactsFollows

Entail

Logical reasoning should ensure that the new configurations represent aspects of the world that actually follow from the aspects that the old configurations represent.

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Entailment

KB: breeze in [2,1]α1 : [1,2] is safe

In every model in which KB is True, α1 is also True, therefore:

KB╞ α1

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Entailment

KB: breeze in [2,1]α2 : [2,2] is safe

There are models in which KB is True, and α2 is not True, therefore:

KB ╞ α2

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• The previous examples not only illustrate entailment, but also shows how entailment can be applied to derive conclusions or to carry out logical inference

• The inference algorithm we used, is called Model Checking • Model checking: Enumerates all possible models to check

that α is true in all models in which KB is true.

Mathematical notation: KB α

Where: i is the inference algorithm used. The notation says: α is derived from KB by i or i derives α from KB.

i

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• To understand entailment and inference:– All consequence of KB is a

haystack– α is a needle

• Entailment: The needle in the haystack• Inference:

Finding the needle

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• An inference procedure can do two things:– Given KB, generate new sentence purported to be

entailed by KB.– Given KB and , report whether or not is entailed by KB.

• Sound or truth preserving: inference algorithm that derives only entailed sentences.– i is sound if: – Whenever KB├ i α, it is also true that KB α ╞

• Completeness: an inference algorithm is complete, if it can derive any sentence that is entailed.– i is complete if: – Whenever KB α , it is also true that KB╞ ├ i α

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Explaining more Soundness and completeness

• Soundness: if the system proves that something is true, then it really is true. The system doesn’t derive contradictions

• Completeness: if something is really true, it can be proven using the system. The system can be used to derive all the true mathematical statements one by one

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Knowledge Representation & ReasoningPropositional Logic

Propositional logic is the simplest logic.• Syntax

• Semantic

• Entailment

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Syntax: It defines the allowable sentences.1. Atomic sentence:

– single proposition symbol.• uppercase names for symbols must have some

mnemonic value: example W1,3 to say the Wumpus is in [1,3].

– True and False: proposition symbols with fixed meaning.2. Complex sentences:

– they are constructed from simpler sentences using logical connectives.

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Logical connectives: • (NOT) negation.• (AND) conjunction, operands are conjuncts.• (OR), operands are disjuncts.• ⇒ (Implication or conditional) It is also known as rule or if-

then statement.• if and only if (biconditional).

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• Logical constants TRUE and FALSE are sentences.• Proposition symbols P1, P2 etc. are sentences.• Symbols P1 and negated symbols P1 are called literals.• If S is a sentence, S is a sentence (NOT).• If S1 and S2 is a sentence, S1 S2 is a sentence (AND).• If S1 and S2 is a sentence, S1 S2 is a sentence (OR).• If S1 and S2 is a sentence, S1 S2 is a sentence (Implies).• If S1 and S2 is a sentence, S1 S2 is a sentence (Equivalent).

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A BNF(Backus-Naur Form) grammar of sentences in propositional Logic is defined by the following rules:

Sentence → AtomicSentence │ComplexSentence AtomicSentence → True │ False │ Symbol

Symbol → P │ Q │ R … ComplexSentence → Sentence

│(Sentence Sentence)│(Sentence Sentence)│(Sentence Sentence)│(Sentence Sentence)

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Order of precedenceFrom highest to lowest:

parenthesis ( Sentence )– NOT – AND – OR – Implies – Equivalent

• Special cases: A B C no parentheses are needed• What about A B C???

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Most sentences are sometimes true. P Q

Some sentences are always true (valid). P P

Some sentences are never true (unsatisfiable). P P

P Q P P Q P Q P Q P Q False False True False False True TrueFalse True True False True True FalseTrue False False False True False FalseTrue True False True True True True4

Mod

els

Sentences

P Q is True in 3 models: P = F & Q =TP = T & Q = FP = T & Q = T

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Implication: P Q

“If P is True, then Q is true; otherwise I’m making no claims about the truth of Q.” (Or: P Q is equivalent to Q)

Under this definition, the following statement is true

Pigs_fly Everyone_gets_an_A

Since “Pigs_Fly” is false, the statement is true irrespective of the truth of “Everyone_gets_an_A”. [Or is it? Correct inference only when “Pigs_Fly” is known to be false.]

P Q P QFalse False TrueFalse True TrueTrue False FalseTrue True True

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Propositional Inference:

Using Enumeration Method (Model checking)

• Let and KB =( C) B C)• Is it the case that KB

╞ ?• Check all possible

models -- must be true whenever KB is true.

A B CKB

( C) B C)

False False False False False

False False True False False

False True False False True

False True True True True

True False False True True

True False True False True

True True False True True

True True True True True

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A B CKB

( C) B C)









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A B CKB

( C) B C)









KB ╞ α

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Propositional Logic: Proof methods

1. Model checking– Truth table enumeration (sound and complete for

propositional logic).– For n symbols, the time complexity is O(2n).– Need a smarter way to do inference

2. Application of inference rules– Legitimate (sound) generation of new sentences from old.– Proof = a sequence of inference rule applications. Can use inference rules as operators in a standard search

algorithm. How?

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Some related concepts: Validity and Satisfiability• A sentence is valid (a tautology) if it is true in all models

e.g., True, A ¬A, A A, • Validity is connected to inference via the Deduction Theorem:

KB α if and only if (╞ KB α) is valid• A sentence is satisfiable if it is true in some model

e.g., A B• A sentence is unsatisfiable if it is false in all models

e.g., A ¬A• Satisfiability is connected to inference via the following:

KB α if and only if (╞ KB ¬α) is unsatisfiable(there is no model for which KB=true and α is false)

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Propositional Logic: Inference rules

An inference rule is sound if the conclusion is true in all cases where the premises are true.

Premise_____ Conclusion

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Propositional Logic: An inference rule: Modus Ponens

• From an implication and the premise of the implication, you can infer the conclusion.

Premise___________ Conclusion

Example:“raining implies soggy courts”, “raining”Infer: “soggy courts”

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Propositional Logic: An inference rule: Modus Tollens

• From an implication and the premise of the implication, you can infer the conclusion.

¬ Premise___________ ¬ Conclusion

Example:“raining implies soggy courts”, “courts not soggy”Infer: “not raining”

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Propositional Logic: An inference rule: AND elimination

• From a conjunction, you can infer any of the conjuncts.

1 2 … n Premise_______________

i Conclusion

• Question: show that ‘Modus Ponens’ and ‘And’ Elimination are sound?

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Propositional Logic: other inference rules

• And-Introduction 1, 2, …, n Premise_______________

1 2 … n Conclusion

• Double Negation Premise

_______ Conclusion

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Propositional Logic: Equivalence rules• Two sentences are logically equivalent iff they are true in the same

models: α ≡ ß iff α β and β α.╞ ╞

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Inference in Wumpus World Let Si,j be true if there is a stench in cell i,j Let Bi,j be true if there is a breeze in cell i,j Let Wi,j be true if there is a Wumpus in cell i,jGiven:1. ¬B1,1 observation 2. B1,1 (P1,2 P2,1) from Wumpus World rulesLet’s make some inferences:

1. (B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1 ) (By definition of the biconditional)

2. (P1,2 P2,1) B1,1 (And-elimination)3. ¬B1,1 ¬(P1,2 P2,1) (equivalence with contrapositive)4. ¬(P1,2 P2,1) (modus ponens)5. ¬P1,2 ¬P2,1 (DeMorgan’s rule)6. ¬P1,2 , ¬ P2,1(And Elimination)

2

1 OK1 2

OK

OK?

?

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Knowledge Representation & Reasoning Inference in Wumpus World

Percept SentencesS1,1 B1,1

S2,1 B1,2

S1,2 B2,1

…

Environment KnowledgeR1: S1,1 W1,1 W2,1 W1,2

R2: S2,1 W1,1 W2,1 W2,2 W3,1

R3: B1,1 P1,1 P2,1 P1,2

R5: B1,2 P1,1 P1,2 P2,2 P1,3

...

Some inferences:Apply Modus Ponens to R1

Add to KBW1,1 W2,1 W1,2

Apply to this AND-EliminationAdd to KB

W1,1

W2,1

W1,2

Initial KB

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Knowledge Representation & Reasoning Inference in Wumpus World

B1,2 P1,1 P2,2 P1,3 (From Wumpus world rules)(B1,2 P1,1 P2,2P1,3 ) (P1,1 P2,2 P1,3 B1,2) (Biconditional

elimination) (P1,1 P2,2 P1,3 B1,2) (And-Elimination) ¬B1,2 ¬ (P1,1 P2,2 P1,3 ) (Contraposition) ¬B1,2 ¬ P1,1 ¬ P2,2 ¬ P1,3 (De Morgan) • Recall that when we were at (1,2) we could not decide

on a safe move, so we backtracked, and explored (2,1), which yielded ¬B1,2

Using (¬B1,2 ¬P1,1 ¬P1,3 ¬P2,2 ) & ¬B1,2 this yields to:¬P1,1 ¬P1,3 ¬P2,2 (Modes Ponens) ¬P1,1 , ¬P1,3 , ¬P2,2 (And-Elimination)

• Now we can consider the implications of B2,1.

3 W?

2S P?

W?1 OK B P?

1 2 3

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1. B2,1 (P1,1 P2,2 P3,1)2. B2,1 (P1,1 P2,2 P3,1)(P1,1 P2,2 P3,1) B2,1 (biconditional Elimination)3. B2,1 (P1,1 P2,2 P3,1) (And-Elimination) + B2,14. P1,1 P2,2 P3,1 (modus ponens)5. P1,1 P3,1 (resolution rule because no pit in (2,2)

¬ P2,2)6. P3,1 (resolution rule because no pit in (1,1) ¬

P1,1)

• The resolution rule: if there is a pit in (1,1) or (3,1), and it’s not in (1,1), then it’s in (3,1).

P1,1 P3,1, ¬P1,1

P3,1

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Resolution

Unit Resolution inference rule:l1 … lk, m

l1 … li-1 li+1 … lk

where li and m are complementary literals.

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Resolution

Full resolution inference rule:l1 … lk, m1 … mn

l1 … li-1li+1 …lkm1…mj-1mj+1... mn

where li and m are complementary literals.

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ResolutionFor simplicity let’s consider clauses of length two:

l1 l2, ¬l2 l3

l1 l3

To derive the soundness of resolution consider the values l2 can take:• If l2 is True, then since we know that ¬l2 l3 holds,

it must be the case that l3 is True.• If l2 is False, then since we know that l1 l2 holds,

it must be the case that l1 is True.

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Resolution1. Properties of the resolution rule:

• Sound• Complete (yields to a complete inference algorithm).

2. The resolution rule forms the basis for a family of complete inference algorithms.

3. Resolution rule is used to either confirm or refute a sentence but it cannot be used to enumerate true sentences.

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Resolution

4. Resolution can be applied only to disjunctions of literals. How can it lead to a complete inference procedure for all propositional logic?

5. Turns out any knowledge base can be expressed as a conjunction of disjunctions (conjunctive normal form, CNF).

E.g., (A ¬B) (B ¬C ¬D)

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Resolution: Inference procedure

6. Inference procedures based on resolution work by using the principle of proof by contradiction:

To show that KB ╞ α we show that (KB ¬α) is unsatisfiable

The process: 1. convert KB ¬α to CNF 2. resolution rule is applied to the resulting clauses.

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Resolution: Inference procedure

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Resolution: Inference procedureExample of proof by contradiction

• KB = (B1,1 (P1,2 P2,1)) ¬ B1,1

• α = ¬P1,2

Question: convert (KB ¬α) to CNF

¬ αKB

Empty Clause

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Inference for Horn clauses. Definite clause: a disjunction of literals of which exactly one is

positive. Example: (L1,1 v ¬Breeze v B1,1) , so all definite clauses are Horn clauses

. Goal clauses: clauses with no positive litterals.

. Horn Form (special form of CNF) KB = conjunction of Horn clauses Horn clause = propositional symbol; or (conjunction of symbols) symbol⇒ e.g., C ( B ⇒ A) (C D ⇒ B)Modus Ponens is a natural way to make inference in Horn KBs

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Inference for Horn clauses

α1, … ,αn, α1 … αn ⇒ β

β

Successive application of modus ponens leads to algorithms that are sound and complete, and run in linear time.

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Inference for Horn clauses: Forward Chaining• Idea: fire any rule whose premises are satisfied in the KB and

add its conclusion to the KB, until query is found.• Forward chaining is sound and complete for horn

knowledge bases

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Inference for Horn clauses: Backward Chaining

• Idea: work backwards from the query q:

check if q is known already, or prove by backward chaining all premises of

some rule concluding q.

Avoid loops:

check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal has already been proved true, or

has already failed.

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Summary• Logical agents apply inference to a knowledge base to derive new information and make

decisions.

• Basic concepts of logic:– Syntax: formal structure of sentences.– Semantics: truth of sentences in possible world or models.– Entailment: necessary truth of one sentence given another.– Inference: process of deriving new sentences from old ones.– Soundness: derivations produce only entailed sentences.– Completeness: derivations can produce all entailed sentences.– Inference rules: patterns of sound inference used to find proofs.– Fwd chaining and Bkw chaining: natural reasoning algorithms for KB in Horn Form.

• Truth table method is sound and complete for propositional logic but Cumbersome in most cases.

• Application of inference rules is another alternative to perform entailment.

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